Finding a mistake in something you wrote is always annoying. When revising the chapter on Atoms and Light for the sixth edition of
Intermediate Physics for Medicine and Biology, I found a whopper. It’s on page 402 of the 5th edition, in the section on Blue and Ultraviolet Radiation. Here is the offending sentence:
The minimum erythemal dose at 254 nm is about 6 × 107 J m-2.
I was trying to add a homework problem to the sixth edition in which I would ask the student to calculate how long it would take to get sunburn for some typical
ultraviolet light intensity and I kept getting a ridiculously long time (years) because our value of 6 × 10
7 is way, way too big. The error goes back to the 3rd edition of
IPMB, where you find a reference for that value:
Diffey, B. L. and Farr, P. M. (1991) Quantitative aspects of ultraviolet erythema. Clin Phys Physiol Meas 12:311-325.
The 3rd edition is even more specific, saying the value is in Table 2 in that paper. So, I obtained the article interlibrary loan (kudos to the Oakland University interlibrary loan office, who got the paper for me in about an hour on a Sunday evening). Here is Diffey and Farr’s Table 2.
The value for minimal erythema at 254 nm is 6 mJ cm-2, which is equivalent to 60 J m-2. I think the incorrect value in IPMB arose because of a unit conversion error. There are 1000 millijoules in a joule, not 1000 joules in a millijoule. Such a mistake would cause a factor of one million error, which would result in an erroneous value of 60 × 106 J m-2, or 6 × 107.
You may have some questions.
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Who did it? Although Russ Hobbie made the initial mistake (he was sole author on the 3rd edition), I read this number when teaching from our book, and when preparing the 4th edition and then again when preparing the 5th edition and never batted an eye. Apparently 60 MJ of UV light causing a person to have only a mild reddening of the skin didn’t bother me at all. I always tell my students to “THINK BEFORE YOU CALCULATE!” but I didn’t.
- If it was in the book, how could it be wrong? Don’t believe everything you read. Just because something is written in a textbook doesn’t make it true. Authors try their best to get everything right, but sometimes they make mistakes. Read critically and thoughtfully. (I’m giving this advice to myself here, more than to you, dear reader).
- What’s erythema? Erythema is redness of the skin. In our context, it is a the initial stages of a sunburn.
- What is the “minimum erythemal dose”? Here’s what Diffey and Farr say: “The erythemal response of the skin to ultraviolet radiation is usually inferred from the minimal erythemal dose (MED). This value is determined by exposing adjacent areas of skin to increasing doses of radiation (usually employing a geometrical series of dose increments) and recording the lowest dose of radiation to achieve erythema at a specified time, usually 24 hours, after irradiation. The visual detection of erythema is subjective and is affected by unrelated factors such as viewing geometry, intensity and spectral composition of ambient illumination, colour of unexposed surrounding skin… and the experience and visual acuity of the observer.” So, it’s the dose where you say “Gosh, my skin is slightly red, I must have gotten a little too much sun today,” and then go about your business with hardly another thought.
- Why did Russ and I give the value for 254 nm? We mean that the ultraviolet radiation has a wavelength of 254 nm, which puts it in the UVC range (100–280 nm). UVC light can certainly cause damage and sunburn, but almost no UVC gets through the earth’s atmosphere to reach our bodies. Most sun tans and sunburns are caused by UVB, which is in a narrow band of wavelengths from 280–315 nm. Wavelengths much shorter are removed by the atmosphere, and the photons for wavelengths much longer do not have enough energy to do significant damage. The wavelength in the above Table 2 that’s most appropriate for this discussion is 300 nm. So, looking at Table 2, perhaps a better value for the minimum erythemal dose would be 24 mJ cm-2 or 240 J m-2. In the sixth edition of IPMB, we will use 200 J m-2 as our typical value (unless we change our minds…when it comes to revising a textbook, it ain’t over till it’s over). Warning: this value depends on factors such as your complexion, so don’t take it too seriously. It’s a ballpark estimate. Everyone is a bit different.
- Well, just how much UVB are we exposed to? We can estimate that from Figure 14.28 in IPMB. In the range from about 295 to 315 nm, the average value of the spectral dose is about 10 mW m-2 nm-1. If we multiply by a 20 nm range, we get 200 mW m-2, or about 0.2 W m-2. That value is for the sun straight overhead (noon near the equator with a clear sky). It’s consistent with other values I have found.
- How is all this related to the “UV index” that the weather forecaster talks about? The UV index is a linear scale (not logarithmic like the decibel scale for hearing), and to calculate it you multiply the intensity in W m-2 by 40. So, the value of 0.2 W m-2 that I quoted earlier corresponds to a UV index of 8. Here in southeast Michigan we can reach a UV index of 8 at noon on a cloudless summer day. In January we are at a UV index of about 2. Latitude and time of the year make a big difference, as does time of the day (in the morning and evening, sunlight comes in at an angle and must therefore pass through more atmosphere than at noon).
- So, how long can I stay in the sun before getting sunburn? On the beach in Hawaii during the summer at noon you can reach a UV index of about 12, so the intensity is 0.3 W m-2, which means 0.3 joules per second per square meter. If your minimum erythemal dose is 200 J m-2, then (0.3 J m-2 s-1) t = 200 J m-2, so t = 667 seconds or 11 minutes. That’s the minimum dose. I bet you could go a half hour before suffering from something you would call a serious sunburn. But if you stay out all afternoon surfing at Waikiki, it could be a problem.
- Can’t I protect myself with sunscreen? Yes, the sun protection factor (SPF) is the factor by which the intensity actually reaching your skin is reduced by the sunscreen. If you put on SPF 30 sunscreen there in Hawaii, your time for a minimum erythemal dose goes up from 11 minutes to five and a half hours. Maybe you can get away with all day, since the UV index will be lower in the morning and evening, extending your time. Just make sure it doesn’t get washed off in the water. You may have to reapply it often.
Let me apologize one more time for the bogus value of minimal erythemal dose in the 5th edition of IPMB. I feel bad about it. I sure hope no one used it to justify spending lots of time in a tanning booth. I call those things “cancer booths.” Stay away from them.
I'll be proofreading the 6th edition of IPMB extra carefully. My motto will be: THINK BEFORE YOU WRITE!
